My Girlfriend and the Four-Block Tower Problem

I decided to have my girlfriend complete the Unifix Cube four-block tower problem. I started by verbally giving her the instructions. I told her to create as many towers as possible with only four blocks per tower and any combination of the orange and blue blocks. I also explained that the orientation of the towers must be like that of a chimney with the narrower portion pointing up. She immediately asked if the towers could be all one color, and I responded that they could be.

Then, my girlfriend got to work. She began by forming a pattern of opposites, but she quickly switched to creating a pattern with four orange blocks, followed by three orange blocks on top and one blue block on the bottom, followed by two orange blocks on top and two blue blocks on the bottom, followed by one orange block on top and three blue blocks on the bottom, followed by four blue blocks, followed by three blue blocks on top and one orange block on the bottom, followed by two blue blocks on top and two orange blocks on the bottom, followed by one blue block on top and three orange blocks on the bottom. She then said, “I think I have all of the combinations.” I asked her to prove it. This made her think for a few moments. I gave her a hint that the order of the cubes matters. Her response was very interesting and enlightening. She said, “Your directions were not clear.” While I felt that my instructions were clear, this proves that it is essential to ensure that every student understands a problem. What sounds good in the teacher’s head may not make any sense to the students.

Then, with a puzzled look on her face, my girlfriend began working with the Unifix Cubes again. However, this time she switched back to a pattern of opposites. When asked why she kept switching from one strategy to another, she simply replied that she kept seeing the problem differently. After a few more minutes, my girlfriend claimed that she thought she had found all of the combinations. However, she could not prove it. I told her to continue working and find more. Thus, she went back to using the Unifix Cubes, and she found two more possible towers for a total of sixteen towers. However, once again, she was unable to prove that this was the correct answer.

Then, I tried to get my girlfriend to come up with an algebraic method for solving the four-cube tower problem. She said the answer of sixteen is based on the algebraic expression 42 because there are four cubes, two colors, and four times four equals sixteen. I asked her how she could prove this, and she said that she could try three blocks. She did this and came up with eight combinations. She then decided to try two-block towers, and she found that there were four combinations. This led her to the solution of C^n, with C representing the number of colors and n representing the number of blocks.

Obviously, proving one’s answer to be correct is higher-level thinking, which means that many students will find it challenging. I also know that if an adult struggled with convincing me that her answer was correct then a middle school student or high school student likely will struggle as well. I feel that the best thing to do is use the Socratic Method and ask students questions that lead them to the right answer or the right explanation. However, it is important not to simply give the students the answer because this defeats the purpose, which is to get students to think mathematically. In addition, students are less likely to understand how to solve a problem and retain any concepts learned if the teacher simply tells them the answers.

Based on my experience in class and this observation, I can clearly see that there are many ways to solve a problem like the four-cube tower problem. However, it does appear that people will search for recognizable patterns rather than simply guessing. This obviously is a time saving measure, and it brings some degree of organization to the problem solving process. Interestingly, my girlfriend kept switching from one pattern to another. What this tells me is that depending on what one discovers while working on a problem can dramatically alter the strategy used.

In conclusion, using Unifix Cubes seems to be a great way to get students interested in a problem that requires mathematical thinking. From my experience as a science teacher, I know that students love working with their hands. Plus, if they learn by discovery it is much more likely for the concepts to remain in their long-term memories. Thus, lessons that utilize Unifix Cubes are highly likely to be effective because students get to learn by doing and learn by discovery.